i'm having some doubts/problems implementing a Multi Level Monte Carlo method. The setting is the following, I have generated some samples (approximated solutions of a PDE) using various polynomial degree, 4 5 6 7. Lots of samples at the low level refinement, few samples at degree 7. For sake of simplicity, the "solution" to the PDE is a real number, and i need the mean. As far as i understood, to get the MLMC Estimate i should do :
MLMC Estimate=Mean at Degree 4+(Mean at Degree 5−Mean at Degree 4)+(Mean at Degree 6−Mean at Degree 5)+(Mean at Degree 7−Mean at Degree 6)
this simplify to
MLMC Estimate=Mean at Degree 7
but i expect not to be the same "number", i'm using "format long" in matlab to get the possible divergence from that value. However, testing my code, i'm trying the above with 100 samples at degree 4 and 10 samples at degree 5, and the MLMC estimate is exactly (in format long) the mean at degree 5. Is it because the sample size is very very small? Should i expect the estimate to diverge from the mean using much more samples? Have i missed something in my understanding of MLMC?